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A Semigroup Approach to Nonlinear Lévy Processes

Author

Listed:
  • Denk, Robert

    (Center for Mathematical Economics, Bielefeld University)

  • Kupper, Michael

    (Center for Mathematical Economics, Bielefeld University)

  • Nendel, Max

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We study the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear Lévy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators ($A_\lambda$) $_{\lambda\in \Lambda}$ of linear Lévy processes which guarantees the existence of a nonlinear Lévy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE $\partial_t u=\sup_{\lambda\in \Lambda} A_\lambda u$. The results are illustrated with several examples.

Suggested Citation

  • Denk, Robert & Kupper, Michael & Nendel, Max, 2019. "A Semigroup Approach to Nonlinear Lévy Processes," Center for Mathematical Economics Working Papers 610, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:610
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    File URL: https://pub.uni-bielefeld.de/download/2934230/2934239
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    References listed on IDEAS

    as
    1. Neufeld, Ariel & Nutz, Marcel, 2014. "Measurability of semimartingale characteristics with respect to the probability law," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3819-3845.
    2. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    3. Dolinsky, Yan & Nutz, Marcel & Soner, H. Mete, 2012. "Weak approximation of G-expectations," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 664-675.
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    Cited by:

    1. Max Nendel, 2021. "Markov chains under nonlinear expectation," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 474-507, January.
    2. Martin Larsson & Marvin S. Mueller & Josef Teichmann, 2020. "Stopper-Controller Games embedded in Single-Player Control Problems," Papers 2006.09493, arXiv.org.
    3. Denk, Robert & Kupper, Michael & Nendel, Max, 2019. "Convex Semigroups on Banach Lattices," Center for Mathematical Economics Working Papers 622, Center for Mathematical Economics, Bielefeld University.
    4. Fuhrmann, Sven & Kupper, Michael & Nendel, Max, 2021. "Wasserstein Perturbations of Markovian Transition Semigroups," Center for Mathematical Economics Working Papers 649, Center for Mathematical Economics, Bielefeld University.
    5. Tolulope Fadina & Ariel Neufeld & Thorsten Schmidt, 2018. "Affine processes under parameter uncertainty," Papers 1806.02912, arXiv.org, revised Mar 2019.
    6. Nendel, Max & Röckner, Michael, 2019. "Upper Envelopes of Families of Feller Semigroups and Viscosity Solutions to a Class of Nonlinear Cauchy Problems," Center for Mathematical Economics Working Papers 618, Center for Mathematical Economics, Bielefeld University.

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